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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

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0 replies
jlacosta
Jun 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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[Registration Open!] 2025 Summer Solstice Math Olympiad
SMO_Team   14
N 41 minutes ago by Vivaandax
[center]
IMAGE

Hello everyone!
[left]
We at Solstice Math Olympiads are extremely happy and excited to announce the SSMO, the Summer Solstice Math Olympiad. If you have any questions, please feel free to email us at solsticematholympiads@gmail.com.
[/left]

Signup here and join our discord!

How the Contest Will be Held
Test Information
Test Rules
Schedule
Prizes (~$10000!!!)
Additional Information

SPONSORS
A big thank you to our sponsors for making the 2025 SSMO possible!!!

Platinum Sponsors:

IMAGE



IMAGE


Gold Sponsors:

IMAGE


Silver Sponsors:

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[/center]
14 replies
SMO_Team
Jun 4, 2025
Vivaandax
41 minutes ago
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The angle bisectors are perpendicular/parallel
Entei   1
N an hour ago by RANDOM__USER
Source: Own
In $\triangle{ABC}$, let two altitudes $BE$ and $CF$ meet at the orthocenter $H$. Let the tangents of circle $(ABC)$ from $B$ and $C$ meet at a point $T$. $AT$ meets $EF$ at $X$. $M$ is the midpoint of $BC$. Prove that the angle bisector of $\angle{XHM}$ is perpendicular to the angle bisector of $\angle{BAC}.$

IMAGE
1 reply
Entei
3 hours ago
RANDOM__USER
an hour ago
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Product of injective/surjective functions
WakeUp   7
N 2 hours ago by lpieleanu
Source: Romanian TST 2001
a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function.

b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.
7 replies
WakeUp
Jan 16, 2011
lpieleanu
2 hours ago
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Multiple of power of two.
dgrozev   4
N 2 hours ago by Assassino9931
Source: Bulgarian TST, 2020, p2
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
4 replies
dgrozev
Aug 4, 2020
Assassino9931
2 hours ago
J
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Parallelity and equal angles given, wanted an angle equality
BarisKoyuncu   6
N 2 hours ago by expsaggaf
Source: 2022 Turkey JBMO TST P4
Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that
$$m(\widehat{ACB})=m(\widehat{PCT})$$
6 replies
BarisKoyuncu
Mar 15, 2022
expsaggaf
2 hours ago
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2020 IGO Intermediate P3
turko.arias   13
N 2 hours ago by fe.
Source: 7th Iranian Geometry Olympiad (Intermediate) P3
In acute-angled triangle $ABC$ ($AC > AB$), point $H$ is the orthocenter and point $M$ is the midpoint of the segment $BC$. The median $AM$ intersects the circumcircle of triangle $ABC$ at $X$. The line $CH$ intersects the perpendicular bisector of $BC$ at $E$ and the circumcircle of the triangle $ABC$ again at $F$. Point $J$ lies on circle $\omega$, passing through $X, E,$ and $F$, such that $BCHJ$ is a trapezoid ($CB \parallel HJ$). Prove that $JB$ and $EM$ meet on $\omega$.


Proposed by Alireza Dadgarnia
13 replies
turko.arias
Nov 4, 2020
fe.
2 hours ago
J
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Functional equation involving decimal-place count
saulgodman   0
2 hours ago
Source: Own

Let
\[ S = \left\{\, x \in \mathbb{Q} : x \text{ has a finite decimal expansion} \,\right\}. \]For each \( x \in S \), define
\[ d(x) = \text{the number of digits after the decimal point in the (reduced) decimal form of } x. \]Find all functions \( f\colon S \to \mathbb{Z} \) such that, whenever both \( x+y \in S \) and \( x y \in S \),
\[ f(x) + f(y) = f(x+y) + d(x y). \]
0 replies
saulgodman
2 hours ago
0 replies
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Did you talk to Noga Alon?
pohoatza   36
N 2 hours ago by ezpotd
Source: IMO Shortlist 2006, Combinatorics 3, AIMO 2007, TST 6, P2
Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,\] where the sum is taken over all convex polygons with vertices in $ S$.

Alternative formulation:

Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A -$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A -$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial
\[ P_A(x) = x^{|A|}(1 - x)^{r(A)}. \]
Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) = 1.$

Proposed by Federico Ardila, Colombia
36 replies
pohoatza
Jun 28, 2007
ezpotd
2 hours ago
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sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant   2
N 2 hours ago by vincentwant
Let $p$ be an odd prime. Define $f(n)$ over the positive integers as follows:
$$f(n)=\begin{cases} \sqrt{n}&\text{ if n is a perfect square} \\ n+p&\text{ otherwise} \end{cases}$$
Let $p$ be chosen such that there exists an ordered pair of positive integers $(n,k)$ where $n>1,p\nmid n$ such that $f^k(n)=n$. Prove that there exists at least three integers $i$ such that $1\leq i\leq k$ and $f^i(n)$ is a perfect square.
2 replies
vincentwant
Apr 30, 2025
vincentwant
2 hours ago
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Aslı tries to make the amount of stones at every unit square is equal
AlperenINAN   1
N 3 hours ago by expsaggaf
Source: Turkey JBMO TST 2025 P2
Let $n$ be a positive integer. Aslı and Zehra are playing a game on an $n\times n$ grid. Initially, $10n^2$ stones are placed on some of the unit squares of this grid.

On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.

For which values of $n$, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
1 reply
AlperenINAN
May 11, 2025
expsaggaf
3 hours ago
J
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Arrange positive divisors of n in rectangular table!
cjquines0   44
N 3 hours ago by ezpotd
Source: 2016 IMO Shortlist C2
Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
44 replies
cjquines0
Jul 19, 2017
ezpotd
3 hours ago
J
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An FE. Who woulda thunk it?
nikenissan   121
N 3 hours ago by lpieleanu
Source: 2021 USAJMO Problem 1
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ \[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]
121 replies
nikenissan
Apr 15, 2021
lpieleanu
3 hours ago
J
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MAN IS KID
DrMath   138
N 4 hours ago by ihatemath123
Source: USAMO 2017 P3, Evan Chen
Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$. Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$. Lines $MK$ and $BC$ meet at $S$, and $N$ is the midpoint of $\overline{IS}$. The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$. Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$.

Proposed by Evan Chen
138 replies
DrMath
Apr 19, 2017
ihatemath123
4 hours ago
J
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2n equations
P_Groudon   84
N 4 hours ago by DeathIsAwe
Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:

\begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\ &\vdots & &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}
84 replies
P_Groudon
Apr 15, 2021
DeathIsAwe
4 hours ago
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Preparation for the AIME
tzhang1   5
N Mar 28, 2011 by trigonometry456103
What chapter in the AoPS Volume 1 or volume2 book would help students develop the skills needed to solve problem #1-15 on the AIME 2011 I?
5 replies
tzhang1
Mar 28, 2011
trigonometry456103
Mar 28, 2011
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Preparation for the AIME
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Reveal topic tags
AMC
AIME
AIME I
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G H BBookmark kLocked kLocked NReply
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tzhang1
664 posts
tzhang1
#1 h  Mar 28, 2011, 10:49 PM • 2 Y
Y by Adventure10, Mango247
What chapter in the AoPS Volume 1 or volume2 book would help students develop the skills needed to solve problem #1-15 on the AIME 2011 I?
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VIPMaster
458 posts
VIPMaster
#2 h  Mar 28, 2011, 10:50 PM • 2 Y
Y by Adventure10, Mango247
Well, if your goal is to solve the entire AIME, I think you would need the entirety of both books. Sorry to be of no help at all, but your question is extremely vague and broad.
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trigonometry456103
349 posts
trigonometry456103
#3 h  Mar 28, 2011, 11:02 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
All of it!
:)

All of it might not even be enough......
But certainly improve your score by a significant amount(if used properly)
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Maxima
1012 posts
Maxima
#4 h  Mar 28, 2011, 11:12 PM • 2 Y
Y by Adventure10, Mango247
Well, he refers to 1-15 on the *2011* AIME. It most certainly is not all of it.
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VIPMaster
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VIPMaster
#5 h  Mar 28, 2011, 11:13 PM • 2 Y
Y by Adventure10, Mango247
...but still a substantial portion of it.
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trigonometry456103
349 posts
trigonometry456103
#6 h  Mar 28, 2011, 11:16 PM • 5 Y
Y by john456852, ThisUsernameIsTaken, Adventure10, Mango247, and 1 other user
That's true.
:)
In that case,
VOLUME 1:
Chapters 3, 4, 6, 7, 11-20, 28

VOLUME 2:
Chapters 1-4, 6, 9, 12, 13, 23, 26
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